I can't speak to their tactics, but I can to their operational doctrine. They were actually relatively similar, with both focusing on using an armored spearhead to create breakthroughs, that were then exploited by infantry and follow-up armor. They differ in a key way though, the German focus was on a single Schwerpunkt, or center of gravity, where a breakthrough on that one point would be decisive. This is clearly seen during the invasion of France, where the Ardennes served as the center of gravity, and when the Germans broke through at that point, it lead to the collapse of the entire Allied defensive line. The Russians on the other hand focused on breakthroughs at multiple points of the enemy line, with the immediate goal to destroy the enemy operational reserves, and then for multiple breakthroughs to meet up and form an encirclement. The counteroffensive during the Battle of Moscow is an example of an attempt to apply this doctrine, they succeeded in breaking through the German lines and pushing them back significantly, and a pocket began to develop around Klin, however they failed to complete the envelopment and the 3rd Panzer Army was able to withdraw.

Sources: Soviet Military Operational Art: In Pursuit of Deep Battle by Glantz and Storm of Steel by Habeck. The first is an in depth coverage of the Soviet Deep Battle doctrine, and the second focuses on both Soviet and German development of their armor doctrines.

Perhaps a more detailed question is: how did the relative scarcity and irreplacibility of their tanks affect German tank tactics?